Implementation in Mathematica

This paper is accompanied by an implementation of the described algorithm in Mathematica. The package (Landen.m) containing the algorithm as well as some examples (examples.nb) are available.

Abstract

A new iterative method for high-precision numerical integration of rational
functions on the real line is presented. The algorithm transforms the rational integrand
into a new rational function preserving the integral on the line. The
coefficients of the new function are explicit polynomials in the original
ones. These transformations depend on the degree of the input and the
desired order of the method. Both parameters are arbitrary. The formulas can
be precomputed. Iteration yields an approximation of the desired integral
with \(m\)-th order convergence. Examples illustrating the
automatic generation of these formulas and the numerical behaviour of this
method are given.